Sound amplifier



1 1927 H. s. SATTERLEE scum) AMPLIFIER Filed Sept. 28, 19 23 2 Sheets-Sheet l m m m m z INVENTOR A TTORNEY I Nov. 1', 1927.

1,647,253 H. S. SATTERLEE scum) AMPLIFIER Filed S ept. 28. 1923 2 Sheets-Sheet M s. JMZLQ gwvemtoz WW J 85 I Qua/meg 6 Patented 0v. I 1, 1927.

UNITED STATES HENRY S. SATTEBLEE, NEW Y ORK, N. Y.

SOUND AMPLIFIER.

Application filed September as, 1923. Serial No. 665,258.

, This invention relates to an improved device and method for the amplifying and distributing of sound waves as, for exam: ple, those produced by a vibrating diaphragm or similar sound generators. preferred form of the invention illustrated is particularly adapted to the amplification and distribution of sound waves reproduced for auditory reception, as exemplified in the mechanically operated diaphragm of the phonograph, and in the electro-magnetically operated diaphragm of the telephone receiver, and it is specially applicable to use as a loud speaker in radio telephony.

In the drawings accompanying this specification and of which they form a part, I show:

Fig. 1, a selected form of sound amplifier, or resonator, and which, for descriptive purposes in th1s ..instance, I have denoted by three parts, viz: the straight portion extending from 7 to 2, the central portion extending from the point of initial curvature at 2 around to a point in the general position of 5, and an end portion extending from 5 to the termination of the spiral and the adjoining flare 17. These portions may be continuous to form an integral whole or certain parts may be made separately and cemented, or joined, together in an appropriate manner as may be best suited to the conditions of use and manufacture. In the development of the spiral of the amplifier, as shown in Fig. 1, both the inner and outer borders have a common centre or origin at 1. The broken lines 1-2, 1-3, 14=,'1-5, 16, 1-8, 1-9, 1-10 are radii vectors from the centre 1 to the spiral borders.

. Fig. 2 is a circle; the limiting case of a logarithmic spiral.

Figs. 3 and 4 show the evolution of the spiral and refer particularly to later mathematical computations.

The invention include; two principal elements; a horn, and an enclosing material for the horn. The horn is characterized by.

having ,its curvature especially designed with respect to the volumetric expansion which has been found to be most advantageous for a conical resonator, the theoretical mathematical limitations of the optimum The i to conform to the physical requirements for maximum reflection, that is to say of minimum absorption of sound wave energy, and at-the same time to limit the resonance to that of the air contained in the cavity to the exclusion of any audible resonance produced by the walls of the horn itself. These features are combined to reproduce all proportionate elements of sound exactly as propagated at the source; a decided improvement over other forms of resonators.

One of the main requisites of a resonating tube orhorn, designed for the amplification of sound waves, is that it shall have adequate length to respond sufficiently to the longest wave lengths which it may be desired to reproduce, which wave lengths extend to the lower tones of the musical scale. That is to say, the sound wave length range which may be considered, forpractical purposes, lies between the frequencies of 129 or C, and 20,480 or E the wave lengths corresponding to these frequencies varying from approximately 48 inches to a small fraction of 1 inch. The transmission of so full a range of musical tones necessitates a length of amplifying horn sufficient to conform to certain requirements. For instance, it has been found that a horn less than 36 inches in length, while functioning well for the higher notes, will not give an adequate re sponse to the lower notes of the musical scale, whereas one over 48 inches in length, while showing no apparent change in the functioning for higher notes, gives an entirely satisfactory response to the lower tones; The explanation of this is that the shorter length resonator can only reproduce the over tones of any notes which are below its range. In other words, a note, which has a wave length longer than the horn, is suppressed and the next higher harmonic which is the octave, and other partials are sounded. This obviously leads to a false rendition of the quality of the original source of sound employed and when the shorter and longer horns are compared side by side, using same sound generator, it may be observed that, in addition to the change of quality, there is a definite acoustic sensation of an octave difference in pitch.

, due to the fundamental resonance of the diaphragm of the generator. I a The distortion due to these two resonance factors is very noticeable with resonating tubes having a terminal opening equal or waves.

greater than half of their length. Technically s eaking, an amplifying horn' may be consi ered as enclosing an air column which is the actual resonator of the sound This air resonator functions according to its size and shape and there is an additional factor sometimes noticeable, depending on the material of the horn; this last is an adventitious factor, due to the intrinsic resonance, or vibration, of the walls of the horn itself and may be recognized in the metallic sound which is associated with the ordinary tin megaphone and with most metallic horns. I have found that, to give most satisfactory results as a distortionless sound amplifier, a straight resonating tube should be more than 36, and referably between 48 and 65 inchesin length and should have an outlet of a diameter equal to about onesixth of the slant length of a corresponding straight cone. This would'be equivalent to considerably less than one-sixth of the length of curvature of a cone which is coiled upon itself. For the ratio of base diameter to slant length in a straight cone is more essentially, for present purposes, the ratio of base area to total volume; and when a cone of given length is coiled-into a spiral, the

volume is diminished accordingly, although the slant length function is preserved in the outer course of the spiral. To preserve the optimum ratio of base area to total volume the diameter of the base must therefore be reduced in the spiral. This results in a modification of the base diameter to slant length ratio when dealing with a curved resonaton The problem of so coiling the, resonating tube as to preserve the acoustic features of a straight cone as far as possible and to avoid distortion by phase interference and loss of wave energy by damping factors of reflection, has been the subject of original investigation by me from a mathematical standpoint and of which I treat later.

I have accordingly designed the curves of the tube, or horn, upon a carefully calculated basis to suit the special requirements as out-o lined above and have adopted the principle gent to any radius vector is a constant, shown as angle a in Figs. 2 and 3. This angle determines the ratio of expansion of the spiral. The value of this constant is important for my purpose as determining the relation between the total length of the resonatinv tube and the diameter available for its outl et. I have found that the constant angle a of 80 degrees 5 minutes produced the desired result, but I do not wish to confine myself to this particular angle as any values between 7 5 and 83 degrees or even below and above this might be selected in choosing a curve for different requirements as to length and volume of resonating tube and bulk of the enclosing material which would be best adapted to any selected sound generator.

As will be seen from Fig. 1, this constant angle applies in developing the curve of both inner and outer borders of the tube, or resonator, and the relation of these two curves is a matter for special selection as will be brought out in a later mathematical treatment.

In development, the tube is symmetrical in cross section, taken along the line of any radius vector, and for the purpose of this specification, such section is assumed to be circular, so that the logarithmic function obtains here in the expansion of a circular cross sectional area, although it is evident that a variety of cross sections might be selected and used effectively.

The straight portion of the resonator, 7 to 2 in Figure l,conforms in circumeference at 2 with the spiral portion of the resonator and may be either a cylinder or a conical tube; a truncated cone being preferably shown with the truncated extremity 7 arranged to fit the sound generator. The slant of this truncated cone approximates a tan gent to the spiral tube at the juncture 2, that is, it makes the aforesaid angle or with the radius vector l2 to the outer border at this point.

In rig. 1 dotted lines between the points 15 and points 16 representthelines of propagation of the wave front from the sound generator. As indicated, these are reflected from the curved wallbf the spiral in a manner to make the angles of reflection equal to the angles of incidence upon the tangent to the spiral at the points impact as shown, for example, at points 15 and points 16 in Figure 1.

The inner boundary of the resonating tube at its initial curve is specially designed to give these reflected rays free passage. From the point of juncture with the straight portion 72, up to 8, another logarithmic spiral f lesser angle on than that of the outer spiral impact of the sound waves.

of the resonator are shown in the same plane in order to demonstrate the nature of the curve in a simple manner. Obviously however it is necessary for clearance, to offset the first port-ion of the tube. This may be effected by rotation upon an axis 1112 which is'taken normal to 13 and upon an axis 13-14 which is taken normal to 1-4. A twenty degree shift on each of these axes will give the required clearance.

Beyond the radius vector 18-4, as shown in Figure 1, the inner boundary of the resonator is a logarithmic spiral exactly equal to that forming the outer boundary but with an angle of retardation, as shown in this instance, of approximately 270 degrees. This angle of retardation determines the cross sectional area and is an essential factor in adjusting the volumetric capacity of the resonator. It may therefore be varied to suit requirements. The total angle of revolution of the spiral portion of the resonator, and consequently its length, is governed by the range of the sound to be reproduced and the condition to be met and therefore will also be variable. In the resonator of Figure 1, this angle is shown as 600 degrees.

As previously stated an important feature of my invention is the substance of which the resonator is made, and for which a dense, non-porous material is used. By way of illustration and because it has been satisfactory in practice, I have fixed upon laster of Paris for this purpose, although 1t will be understood that I do not confine myself to this substance alone as many similar gypsum cements and other materials do equally well and any kind of dense material such as metal, synthetic resins or cements other than gypsum cements would be available for special purposes. Gypsum cement, when first cast, is relatively dense owing to the moisture present and in this condition serves well to reflect a maximum of sound wave energy. However, porosity develops with the evaporation of the moisture and the dry gypsum has a tendency to crumble under the physical d To avoid this, the casting, as first made, may be heated to drive ofl the moisture and then impregnated with waxy or resinous material or a mixture of such materials or any other materials which will render it permanently nonporous and give maximum density in a permanent form. The resonator so constructed forms a dense, smooth-walled and relatively cause of the lateral twist which is required to offset the strai ht portion for clearance of the outer convo utions, it has been found advisable to cast in metal that portion of the tube whichextends from 7 to 4 and to imbed it in the plaster of Paris or such material as may be used for the main portion of the resonator. This union forms a dense-walled non-vibrating central core at the commencement of the spiral. In this first portion of the spiral, the physical impact of the sound waves is a maximum but when these waves have reached the more expanded portion of the resonator, as may be approximately indicated at 5 in Figure 1, they have lost a portion of their energy of impact through expansion and the lnfiuence of various damping factors. Consequently the thickness of the resonator wall from this point on need not be as dense or as massive as in the first portion, but the same spiral develop ment is continued to where a final and slig t flare is provided at the termination of the spiral, said flare being shown at 17 in Figure 1. The size of this flare is of particular importance in the finaladjustment of the volumetric ratio.

In designing an air resonator for use with any special type of sound generator, the wave energy production capacity of that generator should be taken'into account. It is obvious that these generators will vary and that more powerful ones are required to fully energize an air resonator of large volumetric capacity than would be the case with resonators of small capacity. In my method of design, a predetermined volumetric range of capacity is chosen and which is suitable to the sound generator to be employed. VVith in this range is selected, by mathematical formulae to be demonstrated later, a logarithmic spiral tube having a total volume of from 15 to 25 per cent less thanthe pre determined total volume.

This reserved volume is then added in th flare, the extreme diameter of which is restricted to not more than a 50 per cent increase over the diameter of the resonator at the termination of the spiral. I restrict the extreme diameter of the flare for the following reasons. The flare for the end correction of parallel wall pipes, originally formulated by Helmholtz, has the ratio ml- R' /2' where R is the lesser radius and R the extreme radius of the flare. This re resents an increase of 41.42 er cent in iameter and is the degree of are usually employed in musical instruments of the conical type.

. Inv the usual type of resonator, or amplifier, the ratio of the flare often runs as high as 400 or 500 per cent which I believe is done in order to give volumetric capacity in the flare rather than to employ a greater length in the amplifier which would otherwise be. necessary. I maintain that it is better to secure volumetric capacity by increased length and to assi n to the flare only a small percentage of this capacity; preferably per cent or less. Theoretical investigation and practical experiment lead me-to believe that, where there is suflicient length in a true logarithmic spiral resonator, no" flare is necessary and need only be resorted to for the incidental purpose of adjusting the total volumetric capacity of the resonator to the energy capacity of the sound generator. My investigations reveal that a flare has very little, if any, distributive value in conjunction with my resonator.

In the following are brought out the formulae usedin the determination ofmy l0garithmicspiral resonator, the mathematical procedure necessary to produce the effects set forth in the preceding description.

The polar equation of a logarithmic spiral may be writtenwhere R is the radius vector, 5 the Napierian base, 0 the polar angle of revolution in radians and a the constant angle formed with the radius vector by the tangent to the spiral. If R be any known radius vector and or known, any other radius vector, R, may be found for any known value of 0; or, any

' unknown value of 0 may be found from known values of R and R by arranging the formula thus I 0cota (1 Figure 2 and Figure 3 represent tangents drawn to radius vectors enclosing an angle 0 of 90 degrees. Figure 2 is a circle; the limiting case of the logarithmic spiral. Figstant and consequently its suplement, B, is constant; also +w=B and +w ll and in the quadrilateral OPTQ, since 0 equals 90 degrees, III must also equal 90 degrees. It follows that 0=u+m' and |p==+ and the ratios I 2 and w (I,

vary inversely with 0.

I In the spiral, g is a constant K, depending upon a being constant, and x is therefore a function of a.

In the circle, Figure 2, a 90 and 10 0) =l5 and the ratio Now in Figure 4, for example, the logarithmic spiral has the value for a of 77.55775 degrees.

From this value of a the angles to and o may be evaluated as follows z-By equation (l) it may be shown that, for a=77.55775 shown in Figure 3) when OP=1, OQ=J2 or 1.4142. Now the ratio radians (that is 90 degrees as By algebraic proof it may be shown that the polar angles of revolution 8 between points of incidence of the reflected rays develop the same ratio successively as has just been shownfor successive values of m that is to say 5 K.

For 0 w K0) and 0'=KU)+K (Kw) =K ((ofKw) 1 therefore 6 x0 or% x. The summation of a series of 0*represeuting the total revolution Likewise a series giving the summation of the angles of reflection may be expressed:

By substituting the proper value for K in the last term of equation (3) the point of inc-idence of the nth, or say the final, reflection may be determined and likewise by substitution for K in equation (5) the value of the nth or final angle of reflection may be found. provided, either that the point of first incidence and first value of 0 are known, or that the point of first incidence and the first angle of incidence w, are known.

The volume of a logarithmic spiral tube of circular vectorial cross section may be determined by the following formula. which is an application of the volumetric theorem of Pappus for solids of revolution to formula (2) In this equation, the expression -3 na represents the length of the centroid or path described by the centre of mass of the figure of revolution, which is a circle; and the length of this axis is comparable to the alti-" tude of a cone.

In computing the volume of our resonator, as shown in Figure 1. the volume of a hypothetical continuation of the spiral from the point of junction with the straight p0rtion of the tube at 2, Figure 1, to the point of origin. 1, is first obtained by this formula and 'is subtracted from the total volume of a complete spiral tube as computed by the same formula. In its place is. added the volume of the conical or cylindrical portion which forms the connection with the generator.

For the purpose of comparison with a straight cone of equal base and slant length the following formulas are used 2 SlIl U- Where U is the sen'ii-vertical angle of a right cone having the base Rr and a slant length equal to R see a Biz" L- or RcosU 2RcosU I claim: I

1. In a sound amplifier, a member having a sound amplifying chamber formed therein the boundaries 0t which are logarithmically developed spiral curves.

2. In a sound amplifier, a member having logarithmic spirally curved walls wherein .the angle formed by any tangent to the curve and a radius vector to the point of tangency is constant and lies between the values of 75 degrees and 83 degrees.

In testimony whereof, I have signed my name to this specification this 21st day of September, 1923.

HENRY s. SATTERLEE.

. t \Vhere R see a cos U equals the altitude Y 

